The Regularised Inertial Dean-Kawasaki equation: discontinuous Galerkin approximation and modelling for low-density regime

TL;DR

This paper develops a Discontinuous Galerkin discretisation for the Regularised Inertial Dean-Kawasaki model, ensuring stability and positivity of solutions, and demonstrates its application to particle systems with open-source code.

Contribution

It introduces a DG scheme for the RIDK model with convergence analysis and modifications to ensure positivity, along with numerical validation and an application to reactive particle systems.

Findings

DG scheme achieves stable, convergent simulations of RIDK

Modified RIDK models preserve positivity of density

Numerical results show physically realistic density profiles

Abstract

The Regularised Inertial Dean-Kawasaki model (RIDK) -- introduced by the authors and J. Zimmer in earlier works -- is a nonlinear stochastic PDE capturing fluctuations around the mean-field limit for large-scale particle systems in both particle density and momentum density. We focus on the following two aspects. Firstly, we set up a Discontinuous Galerkin (DG) discretisation scheme for the RIDK model: we provide suitable definitions of numerical fluxes at the interface of the mesh elements which are consistent with the wave-type nature of the RIDK model and grant stability of the simulations, and we quantify the rate of convergence in mean square to the continuous RIDK model. Secondly, we introduce modifications of the RIDK model in order to preserve positivity of the density (such a feature only holds in a ''high-probability sense'' for the original RIDK model). By means of numerical simulations, we show that the modifications lead to physically realistic and positive density profiles. In one case, subject to additional regularity constraints, we also prove positivity. Finally, we present an application of our methodology to a system of diffusing and reacting particles. Our Python code is available in open-source format.